Codes and Expansions (CodEx) Seminar

Chris Cox (Carnegie Mellon University / Ohio State University)Nearly Orthogonal Vectors

How can $d+k$ vectors in $\mathbb R^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $\theta(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb R^d$. In this work, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $\theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb R^k$. Using this connection, we are able to pin down $\theta(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establish asymptotics for general $k$. The main tool is an upper bound on $\mathbb E_{x,y\sim\mu}|\langle x,y\rangle|$ whenever $\mu$ is an isotropic probability mass on $\mathbb R^k$, which may be of independent interest. Our results translate naturally to the analogous question in $\mathbb C^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $\mathbb C^k$, also known as SIC-POVM inphysics literature. Joint work with Boris Bukh.